Olivier Fruchart SPI SPINTEC, Uni niv. Gr Grenob oble le Al - - PowerPoint PPT Presentation

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Olivier Fruchart SPI SPINTEC, Uni niv. Gr Grenob oble le Al - - PowerPoint PPT Presentation

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Note: part of the lecture was made on the blackboard, and does not appear here Olivier Fruchart SPI SPINTEC, Uni niv. Gr Grenob oble le Al Alpes / CN / CNRS S / / CE CEA-INAC, Fr Fran ance www.spintec.fr email:

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Lecture in Sevilla, Spain

Olivier Fruchart

SPI SPINTEC, Uni

  • niv. Gr

Grenob

  • ble

le Al Alpes / CN / CNRS S / / CE CEA-INAC, Fr Fran ance www.spintec.fr email: olivier.fruchart@cea.fr Slides: http://fruchart.eu/slides

Note: part of the lecture was made on the blackboard, and does not appear here

Possible reading: Lecture notes in nanomagnetism: http://fruchart.eu/olivier/publications

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Olivier FRUCHART Magnetization textures 12th June 2017 Lecture Sevilla, Spain

MOTIVATION – MATERIALS

Numerous and complex shape of domains

Magnetic domains

History: Weiss domains Practical: improve material properties

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Olivier FRUCHART Magnetization textures 12th June 2017 Lecture Sevilla, Spain

MOTIVATION – SPINTRONIC DEVICES

Magnetic bits on hard disk drives Underlying microstructure

Co-based hard disk media : bits 50nm and below

  • B. C. Stipe, Nature Photon. 4, 484 (2010)
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Olivier FRUCHART Magnetization textures 12th June 2017 Lecture Sevilla, Spain

Motivation Basics Statics Dynamics

TABLE OF CONTENTS

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Olivier FRUCHART Magnetization textures 12th June 2017 Lecture Sevilla, Spain

𝑋 = 𝜈0 ΰΆ» 𝐈. d𝐍

BASICS – CONTROL MAGNETIZATION REVERSAL

The hysteresis loop

Magnetization reversal under magnetic field The most widespread characterization Spontaneous β‰  Saturation Spontaneous magnetization Coercive field Remanent magnetization Losses

𝐂 = 𝜈0 𝐈 + 𝐍 𝐊 = 𝜈0𝐍

Magnetic induction Another notation

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Olivier FRUCHART Magnetization textures 12th June 2017 Lecture Sevilla, Spain

Soft magnetic material

BASICS – SOFT AND HARD MAGNETIC MATERIALS

Hard magnetic material

Transformers Magnetic shielding, flux guides Magnetic sensors Magnetic recording Permanent magnets

Material composition and crystal structure Microstructure What determines hysteresis loops?

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Olivier FRUCHART Magnetization textures 12th June 2017 Lecture Sevilla, Spain

BASICS – DOMAINS, FROM BULK TO NANO

Bulk material

Numerous and complex shape of domains FeSi soft magnetic sheet

  • A. Hubert, Magnetic domains

Mesoscopic scale Nanoscopic scale

Small number of domains, simple shape Microfabricated elements Kerr microscopy

  • A. Hubert, Magnetic domains

Magnetic single domain Nanofabricated dots MFM

Sample courtesy: I. Chioar, N.Rougemaille

Nanomagnetism β‰ˆ Mesomagnetism

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Olivier FRUCHART Magnetization textures 12th June 2017 Lecture Sevilla, Spain

Motivation Basics Statics Dynamics

TABLE OF CONTENTS

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Olivier FRUCHART Magnetization textures 12th June 2017 Lecture Sevilla, Spain

STATICS – MICROMAGNETISM FORMALISM

Magnetization

Magnetization vector M

𝐍(𝐬) = 𝑁𝑦 𝑁𝑧 𝑁𝑨 = 𝑁s 𝑛𝑦 𝑛𝑧 𝑛𝑨

Continuous function May vary over time and space Modulus is constant and uniform (hypothesis in micromagnetism)

𝑛𝑦

2 + 𝑛𝑧 2 + 𝑛𝑨 2 = 1

Mean field approach is possible:

𝑁s = 𝑁s π‘ˆ Exchange interaction

Atomistic view

β„° = βˆ’ ෍

π‘—β‰ π‘˜

𝐾𝑗,π‘˜π“π‘—. π“π‘˜

(total energy, J) Micromagnetic view

𝐓𝑗. π“π‘˜ = 𝑇2cos(πœ„π‘—,π‘˜) β‰ˆ 𝑇2 1 βˆ’ πœ„π‘—,π‘˜

2

2 𝐹ex = 𝐡 𝛂. m 2 = 𝐡 ෍

𝑗,π‘˜

πœ–π‘›π‘— πœ–π‘¦π‘˜

2

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Olivier FRUCHART Magnetization textures 12th June 2017 Lecture Sevilla, Spain

STATICS – THE VARIOUS TYPES OF MAGNETIC ENERGY

Exchange energy Magnetocrystalline anisotropy energy Magnetostatic energy Zeeman energy (β†’ enthalpy) 𝐹ex = 𝐡 𝛂. m 2 = 𝐡 ෍

𝑗,π‘˜

πœ–π‘›π‘— πœ–π‘¦π‘˜

2

𝐹mc = 𝐿 𝑔(πœ„, πœ’) 𝐹Z = βˆ’πœˆ0𝐍. 𝐈 𝐹d = βˆ’ 1 2 𝜈0𝐍. 𝐈d

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Olivier FRUCHART Magnetization textures 12th June 2017 Lecture Sevilla, Spain

STATICS – DIPOLAR ENERGY

Analogy with electrostatics div 𝐈d = βˆ’div 𝐍 𝐈d 𝐬 = βˆ’π‘s ΰΆΈ

space

div 𝐧 𝐬′ . (𝐬 βˆ’ 𝐬′) 4𝜌 𝐬 βˆ’ 𝐬′ 3 dπ‘Šβ€² 𝐈d 𝐬 = ΰΆΈ 𝜍 𝐬′ . 𝐬 βˆ’ 𝐬′ 4𝜌 𝐬 βˆ’ 𝐬′ 3 dπ‘Šβ€² + ΰΆΎ 𝜏 𝐬′ . 𝐬 βˆ’ 𝐬′ 4𝜌 𝐬 βˆ’ 𝐬′ 3 d𝑇′

Maxwell equation β†’ To lift the singularity that may arise at boundaries, a volume integration around the boundaries yields:

Magnetic charges 𝜍(r)= βˆ’ 𝑁Sdiv 𝐧(𝐬) 𝜏(r)=𝑁S𝐧 𝐬 . 𝐨(𝐬) Usefull expressions β„°d = βˆ’ 1 2 𝜈0 ΰΆΈ

Sample

𝐍. 𝐈ddπ‘Š β„°d = 1 2 𝜈0 ΰΆΈ

Space

𝐈d

πŸ‘dπ‘Š

Always positive Zero means minimum

Hd depends

  • n shape, not

size Synonym: dipolar, magnetostatic

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Olivier FRUCHART Magnetization textures 12th June 2017 Lecture Sevilla, Spain

STATICS – MAGNETIC CHARGES

Examples of magnetic charges

Note for infinite cylinder: no charge β„° = 0 Charges on side surfaces Surface and volume charges

Dipolar energy favors alignement of magnetization with longest direction of sample Take-away message

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Olivier FRUCHART Magnetization textures 12th June 2017 Lecture Sevilla, Spain

STATICS – DIPOLAR FIELDS

Vocabulary

Generic names Magnetostatic field Dipolar field Inside material Demagnetizing field Oustide material Stray field

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Olivier FRUCHART Magnetization textures 12th June 2017 Lecture Sevilla, Spain

STATICS – TENDENCY FOR FLUX-CLOSURE DOMAINS

Films with easy axis out-of-the-plane: Kittel domains

  • C. Kittel, Physical theory of ferromagnetic domains, Rev. Mod. Phys. 21, 541 (1949)

Principle: compromise between gain in dipolar energy, and cost in wall energy

  • H. A. M. van den Berg, J. Magn. Magn. Mater. 44, 207 (1984)

Principle: Reduce dipolar energy to zero

Nanostructures with in-plane magnetization Van den Berg theorem

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Olivier FRUCHART Magnetization textures 12th June 2017 Lecture Sevilla, Spain

STATICS – MAGNETIC LENGTH SCALES

The dipolar exchange length J/m J/m3 Ξ”u = 𝐡/𝐿 Ξ”u ≃ 1 nm β†’ 100 nm

Hard Soft

The anisotropy exchange length

When: anisotropy and exchange compete When: anisotropy and exchange compete

𝐿d = 1 2 𝜈0𝑁s

2

J/m J/m3 Ξ”d = 𝐡/𝐿d = 2𝐡/𝜈0𝑁s

2

Ξ”d ≃ 3 βˆ’ 10 nm

Critical single-domain size, relevant for small particles made of soft magnetic materials Often called: exchange length Sometimes called: Bloch parameter, or wall width

Other length scales can be defined, e.g. with magnetic field

Exchange

Note:

Dipolar

𝐹 = 𝐡 πœ–π‘›π‘— πœ–π‘¦π‘˜

2

+ 𝐿d sin2 πœ„

Exchange Anisotropy

𝐹 = 𝐡 πœ–π‘›π‘— πœ–π‘¦π‘˜

2

+ 𝐿 sin2 πœ„

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Olivier FRUCHART Magnetization textures 12th June 2017 Lecture Sevilla, Spain

STATICS – DOMAIN WALLS AND DIMENSIONALITY

Bloch wall in the bulk (2D)

No magnetostatic energy Width Energy

Ξ”u = 𝐡/𝐿 𝛿w = 4 𝐡𝐿

Other angles & anisotropy

  • F. Bloch, Z. Phys. 74, 295 (1932)

Domain walls in thin films (towards 1D)

Bloch wall NΓ©el wall

𝑒 ≳ π‘₯ 𝑒 ≲ π‘₯

Implies magnetostatic energy No exact analytic solution

  • L. NΓ©el, C. R. Acad. Sciences 241, 533 (1956)

Constrained walls (eg in strips)

Permalloy (15nm) Strip width 500nm

Vortex (1D β†’ 0D)

  • T. Shinjo et al.,

Science 289, 930 (2000)

Bloch point (0D)

Point with vanishing magnetization

  • W. DΓΆring,

JAP 39, 1006 (1968)

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Olivier FRUCHART Magnetization textures 12th June 2017 Lecture Sevilla, Spain

STATICS – WALLS AND TOPOLOGY (BLOCH POINT)

What is a Bloch point?

A magnetization texture with local cancellation of the magnetization vector

  • R. Feldkeller,
  • Z. Angew. Physik 19, 530 (1965)
  • W. DΓΆring,
  • J. Appl. Phys. 39, 1006 (1968)

Bloch-point wall, theory 𝐸 ≳ 7𝛦d

2

  • H. Forster et al., J. Appl. Phys. 91, 6914 (2002)
  • A. Thiaville, Y Nakatani, Spin dynamics in confined

magnetic structures III, 101, 161-206 (2006)

Bloch-point wall, experiments

Experiment Simulation WIRE SHADOW Shadow XMCD-PEEM

  • S. Da-Col et al., PRB (R) 89, 180405, (2014)
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Olivier FRUCHART Magnetization textures 12th June 2017 Lecture Sevilla, Spain

Claims and facts

STATIC – WALLS AND TOPOLOGY (SKYRMIONS)

The Dzyaloshinskii-Moriya interaction

Usual magnetic exchange

ℰ𝑗,π‘˜ = βˆ’πΎπ‘—,π‘˜π“π‘—. π“π‘˜ β„°DMI = βˆ’πžπ‘—,π‘˜. 𝐓𝑗 Γ— π“π‘˜

Promotes ferromagnetism (or antiferromagnetism) The DM interaction Promotes spirals and cycloids

Magnetic skyrmions

  • I. Dzyaloshiinsky, J. of Phys. Chem. Solids 4,

241 (1958)

  • T. Moriya, Phys. Rev. 120, 91 (1960)

A.Fert and P.M.Levy, PRL 44, 1538 (1980)

Requires: loss of inversion symmetry Bulk FeCoSi Lorentz microscopy 90 nm

  • O. Boulle et al.,
  • Nat. Nanotech.,

11, 449 (2016)

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Olivier FRUCHART Magnetization textures 12th June 2017 Lecture Sevilla, Spain

Motivation Basics Statics Dynamics

TABLE OF CONTENTS

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Olivier FRUCHART Magnetization textures 12th June 2017 Lecture Sevilla, Spain

Pioneering experiment of precessional magnetization reversal

DYNAMICS – THE LANDAU-LIFSHITZ EQUATION

LLG equation

Describes: precessional dynamics of magnetic moments Applies to magnetization, with phenomenological damping

  • C. Back et al., Science 285, 864 (1999)

Gyromagnetic ratio

d𝐧 d𝑒 = 𝛿0𝐧 Γ— 𝐈 + 𝛽𝐧 Γ— d𝐧 d𝑒 𝛿0 = 𝜈0𝛿 < 𝟏 𝛿𝑑 = 28 GHz/T 𝛽 > 0 Damping

coefficient

𝛽 = 0.1 βˆ’ 0.001

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Olivier FRUCHART Magnetization textures 12th June 2017 Lecture Sevilla, Spain

DYNAMICS – MOTION OF DOMAIN WALLS

d𝐧 d𝑒 = 𝛿0𝐧 Γ— 𝐈 + 𝛽𝐧 Γ— d𝐧 d𝑒 Precessional dynamics under field

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Olivier FRUCHART Magnetization textures 12th June 2017 Lecture Sevilla, Spain

DYNAMICS – SPIN TRANSFER PHENOMENA

Macrospins (1d model) d𝐍2 d𝑒 = 𝛿0𝐍2 Γ— 𝐈eff + 𝛽 𝐍2 𝑁s,2 Γ— d𝐍2 d𝑒 βˆ’ 𝑄trans𝐍2 𝐍2 Γ— 𝐍1

  • J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996)
  • L. Berger, Phys. Rev. B 54, 9353 (1996)

Magnetization texture (domain wall etc.) d𝐧 d𝑒 = 𝛿0𝐧 Γ— 𝐈 + 𝛽𝐧 Γ— d𝐧 d𝑒 βˆ’ 𝐯. 𝛂 𝐧 + 𝛾𝐧 Γ— 𝐯. 𝛂 𝐧

  • A. Thiaville, Y. Nakatani, Micromagnetic simulation of domain wall dynamics in nanostrips, in

Nanomagnetism and Spintronics, Elsevier (2009)

𝑄trans~𝑄 𝐾 |𝑓|

Number of spin-polarized per unit time Transfer Field-like

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Olivier FRUCHART Magnetization textures 12th June 2017 Lecture Sevilla, Spain

DYNAMICS – MOTION OF DOMAIN WALLS

Precessional dynamics under current d𝐧 d𝑒 = 𝛿0𝐧 Γ— 𝐈 + 𝛽𝐧 Γ— d𝐧 d𝑒 βˆ’ 𝐯. 𝛂 𝐧 + 𝛾𝐧 Γ— 𝐯. 𝛂 𝐧

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Lecture in Sevilla, Spain www.spintec.fr email: olivier.fruchart@cea.fr Slides: http://fruchart.eu/slides